Variations on Dirichlet's theorem

Lior Fishman, David Simmons*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We give a necessary and sufficient condition for the following property of an integer d∈N and a pair (a,A)∈R2: There exist κ>0 and Q0∈N such that for all x∈Rd and Q≥Q0, there exists p/q∈Qd such that 1≤q≤Q and x-p/q≤κq-aQ-A. This generalizes Dirichlet's theorem, which states that this property holds (with κ=Q0=1) when a=1 and A=1/d. We also analyze the set of exceptions in those cases where the statement does not hold, showing that they form a comeager set. This is also true if Rd is replaced by an appropriate "Diophantine space", such as a nonsingular rational quadratic hypersurface which contains rational points. Finally, in the case d=1 we describe the set of exceptions in terms of classical Diophantine conditions.

Original languageEnglish
Pages (from-to)11-22
Number of pages12
JournalJournal of Number Theory
Volume162
Early online date8 Dec 2015
DOIs
Publication statusPublished - 1 May 2016

Bibliographical note

© 2015 Elsevier Inc. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy.

Keywords

  • Diophantine approximation
  • Dirichlet's theorem

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